Problem 69 - A Graphical Approach to Precalculus with Limits

Question

Give the focus, directrix, and axis of each parabola. $$x^{2}=-\frac{1}{2} y$$

Step-by-Step Solution

Answer

Focus: (0, -1/8), Directrix: y = 1/8, Axis: x = 0.

Step 1: Identify the Parabola Form

The given equation is \(x^{2}=-\frac{1}{2} y\). This equation can be rewritten in the standard form of a vertical parabola: \(x^2 = 4py\), where \(p\) is the distance from the vertex to the focus and from the vertex to the directrix.

Step 2: Match Equation with Standard Form

Rewrite the equation in the form \(x^2 = 4py\): \[x^2 = -\frac{1}{2} y.\] Compare this with \(x^2 = 4py\). Here, \(4p = -\frac{1}{2},\) so \(p = -\frac{1}{8}.\)

Step 3: Determine the Focus

For a parabola \(x^2 = 4py\), the vertex of the parabola is at the origin \((0, 0)\). Given \(p = -\frac{1}{8},\) the focus is at \((0, p) = (0, -\frac{1}{8})\).

Step 4: Find the Directrix

The directrix of a parabola \(x^2 = 4py\) is the line \(y = -p\). Since \(p = -\frac{1}{8},\) the directrix is the line \(y = \frac{1}{8}.\)

Step 5: Identify the Axis of the Parabola

The axis of a parabola \(x^2 = 4py\) is the vertical line \(x = 0\). This is because the parabola opens up or down along the y-axis.

Key Concepts

Focus of a Parabola

The focus is a special point inside the parabola that helps define its shape and directly influences how the parabola is oriented. In the case of a vertical parabola given by an equation like \(x^2 = 4py\), the focus is placed somewhere along the y-axis.
This point is crucial because every point on the parabola is equidistant from the focus and a line called the directrix.

To determine the focus, you start by looking at the value of \(p\). For our specific equation \(x^2 = -\frac{1}{2} y\), we find that \(p = -\frac{1}{8}\).
This means that the focus is at the coordinates \((0, -\frac{1}{8})\). The negative sign tells us that the focus is below the vertex of the parabola, which is located at the origin \((0, 0)\). This orientation indicates that the parabola opens downward.

Directrix of a Parabola

The directrix is a straight line that works together with the focus to define the parabola. For any point on the parabola, the distance to the focus is equal to the distance to the directrix. It helps in understanding the geometric nature of a parabola.
For a standard vertical parabola given by \(x^2 = 4py\), the directrix is a horizontal line given by the equation \(y = -p\). This is directly tied to how the value of \(p\) affects the orientation of the parabola.

In our specific equation, where \(x^2 = -\frac{1}{2} y\), we identified \(p = -\frac{1}{8}\).

• The directrix becomes \(y = \frac{1}{8}\).

This means the directrix is above the vertex at the origin, further confirming the downward opening of the parabola. It provides a symmetric balance to the focus placed below the vertex.

Axis of a Parabola

The axis of a parabola is a critical concept as it refers to the line that runs through the vertex and is perpendicular to the directrix. It serves as the line of symmetry for the parabola, meaning that the parabola is a mirror image on either side of the axis.

In a vertically oriented parabola like the one given by \(x^2 = 4py\), this axis is typically the y-axis. This ensures that as the parabola opens upward or downward, both sides are perfectly symmetrical.

For our example, with the equation \(x^2 = -\frac{1}{2} y\), the axis of the parabola is the line \(x = 0\).
This is simply the y-axis, supporting the understanding that the vertex is at the origin and the parabola extends symmetrically along this line.